Divergence Phase Index (DPI) Overview

• DPI is a framework that quantifies phase differences in signals by leveraging Hilbert and Riesz transforms to capture structural changes.
• It is invariant to intensity scaling and rotation, making it effective for comparing both 1D signals and multidimensional images.
• Experimental results highlight DPI's utility in diverse applications such as iEEG analysis, microscopy, and art image studies with significant statistical validation.

The Divergence Phase Index (DPI) is a framework for quantifying phase differences in one and multidimensional signals, grounded in harmonic analysis via the Riesz transform. Introduced as an extension of classical Hilbert Transform phase measures, it defines a geometry-aware phase-difference metric that is invariant to intensity scaling and sensitive to structural changes. The formulation is given for 1D signals through Hilbert-derived instantaneous phase, and for $n$-dimensional fields through componentwise phase variables built from the Riesz transform. Reported applications include intracranial EEG (iEEG) recordings during epileptic seizures, high-resolution microscopy images, and paintings; in these settings, DPI is described as robust to amplitude variation, responsive to structural modifications, and capable of detecting rotational variations in highly isotropic microscopy images (Catanzariti et al., 6 Oct 2025).

1. One-dimensional definition

In the 1D setting, DPI is built from the Hilbert transform and the analytic signal. For a real-valued signal $f(t)$ on $\mathbb R$, the Hilbert transform $Hf$ is defined in the Fourier domain by

$\widehat{Hf}(\xi)\;=\;-\,i\,\sgn(\xi)\,\widehat f(\xi),$

where

$$\widehat f(\xi)=\int_{\mathbb R}f(t)e^{-2\pi i\,\xi t}\,dt$$

and $\sgn(\xi)=+1$ for $\xi>0$, $-1$ for $\xi<0$.

The associated analytic signal is

$f(t)$0

Its amplitude and phase are

$f(t)$1

Given two 1D signals $f(t)$2 and $f(t)$3, the point-wise phase difference is defined by

$f(t)$4

The Divergence Phase Index between $f(t)$5 and $f(t)$6 over a time interval $f(t)$7, or over $f(t)$8 discrete samples, is the average phase difference

$f(t)$9

This construction places DPI in direct continuity with classical Hilbert-phase analysis while replacing event-level or synchronization-specific summaries with an average point-wise phase divergence.

2. Multidimensional extension through the Riesz transform

The multidimensional generalization replaces the Hilbert transform with the Riesz transform. For $\mathbb R$0, the $\mathbb R$1-th Riesz transform $\mathbb R$2 is defined in the Fourier domain by

$\mathbb R$3

Collecting the components yields the vector operator

$\mathbb R$4

For $\mathbb R$5, the $\mathbb R$6-th phase component of $\mathbb R$7 is

$\mathbb R$8

Given two $\mathbb R$9-dimensional fields $Hf$0 and $Hf$1, the phase-difference vector is

$Hf$2

with norm

$Hf$3

The global DPI is the spatial average over a domain $Hf$4:

$Hf$5

Within this formulation, DPI is not restricted to 2D imagery. The paper states that the construction extends mathematically to arbitrary dimension via Riesz transforms, which is the basis for its positioning as a multidimensional phase-difference framework rather than a specialized image-comparison heuristic.

3. Geometric interpretation and invariance properties

Several geometric properties are explicit in the formulation. First, the local phase vector $Hf$6 encodes directional structure, analogously to gradient orientation, without depending on amplitude. Second, because each $Hf$7 is linear and homogeneous of degree zero in amplitude, DPI is invariant under scalar intensity changes: if $Hf$8, then $Hf$9. The statement is also given in the equivalent form $\widehat{Hf}(\xi)\;=\;-\,i\,\sgn(\xi)\,\widehat f(\xi),$0 for any $\widehat{Hf}(\xi)\;=\;-\,i\,\sgn(\xi)\,\widehat f(\xi),$1.

A further property is rotation covariance. Using commutation of Fourier and orthogonal rotations $\widehat{Hf}(\xi)\;=\;-\,i\,\sgn(\xi)\,\widehat f(\xi),$2, the Riesz vector obeys

$\widehat{Hf}(\xi)\;=\;-\,i\,\sgn(\xi)\,\widehat f(\xi),$3

Accordingly, under $\widehat{Hf}(\xi)\;=\;-\,i\,\sgn(\xi)\,\widehat f(\xi),$4, the phase vector rotates by $\widehat{Hf}(\xi)\;=\;-\,i\,\sgn(\xi)\,\widehat f(\xi),$5. This property is central to the rotation-detection experiment on isotropic microscopy images.

DPI is also stated to be sensitive to structural differences. If $\widehat{Hf}(\xi)\;=\;-\,i\,\sgn(\xi)\,\widehat f(\xi),$6 differs from $\widehat{Hf}(\xi)\;=\;-\,i\,\sgn(\xi)\,\widehat f(\xi),$7 only by shape changes such as edges or texture, then DPI detects nonzero $\widehat{Hf}(\xi)\;=\;-\,i\,\sgn(\xi)\,\widehat f(\xi),$8, even when intensities match. Conversely, uniform intensity scaling alone does not alter the phase components. This combination of sensitivity and invariance is the core geometric distinction of DPI relative to amplitude-dependent comparison schemes. A plausible implication is that DPI is most informative when the operative variable of interest is structural organization rather than absolute signal magnitude.

4. Computational procedure and asymptotic cost

The 1D computation takes as input signals $\widehat{Hf}(\xi)\;=\;-\,i\,\sgn(\xi)\,\widehat f(\xi),$9, $$\widehat f(\xi)=\int_{\mathbb R}f(t)e^{-2\pi i\,\xi t}\,dt$$0, and sampling $$\widehat f(\xi)=\int_{\mathbb R}f(t)e^{-2\pi i\,\xi t}\,dt$$1, with an optional bandpass filter, and returns $$\widehat f(\xi)=\int_{\mathbb R}f(t)e^{-2\pi i\,\xi t}\,dt$$2. The stated procedure is: optionally bandpass-filter $$\widehat f(\xi)=\int_{\mathbb R}f(t)e^{-2\pi i\,\xi t}\,dt$$3 to narrowband; compute FFTs $$\widehat f(\xi)=\int_{\mathbb R}f(t)e^{-2\pi i\,\xi t}\,dt$$4, $$\widehat f(\xi)=\int_{\mathbb R}f(t)e^{-2\pi i\,\xi t}\,dt$$5; form Hilbert multipliers $$\widehat f(\xi)=\int_{\mathbb R}f(t)e^{-2\pi i\,\xi t}\,dt$$6; compute $$\widehat f(\xi)=\int_{\mathbb R}f(t)e^{-2\pi i\,\xi t}\,dt$$7, $$\widehat f(\xi)=\int_{\mathbb R}f(t)e^{-2\pi i\,\xi t}\,dt$$8; apply inverse FFT to obtain $$\widehat f(\xi)=\int_{\mathbb R}f(t)e^{-2\pi i\,\xi t}\,dt$$9 and $\sgn(\xi)=+1$0; then for each sample $\sgn(\xi)=+1$1 evaluate $\sgn(\xi)=+1$2, $\sgn(\xi)=+1$3, and $\sgn(\xi)=+1$4 wrapped to $\sgn(\xi)=+1$5; finally return $\sgn(\xi)=+1$6. The complexity is dominated by two FFTs and two inverse FFTs, giving $\sgn(\xi)=+1$7.

For images, the 2D procedure takes images $\sgn(\xi)=+1$8 of size $\sgn(\xi)=+1$9, partitioned into $\xi>0$0 non-overlapping square patches $\xi>0$1. The steps are: compute $\xi>0$2 of $\xi>0$3 and $\xi>0$4; for each frequency $\xi>0$5, build Riesz kernels $\xi>0$6, $\xi>0$7; compute the transformed components $\xi>0$8, $\xi>0$9, and the corresponding quantities for $-1$0; invert to obtain $-1$1; define the pointwise phase fields $-1$2, $-1$3, where $-1$4, and analogously for $-1$5; at each patch $-1$6, compute pixelwise $-1$7 and average over the patch to obtain $-1$8; optionally binarize $-1$9 via the “elbow” method to highlight structural changes. The complexity is $\xi<0$0 for 2D FFTs plus $\xi<0$1 pointwise operations (Catanzariti et al., 6 Oct 2025).

The workflow indicates that the method is fundamentally spectral, with local spatial summarization introduced at the patch level rather than by replacing the global Fourier-domain construction.

5. Experimental results

The 1D experiment concerns iEEG during epileptic seizure. The dataset is a 9-channel intracranial EEG recording sampled at $\xi<0$2 Hz over $\xi<0$3 s total, divided into $\xi<0$4 s interictal and $\xi<0$5 s ictal, with a $\xi<0$6 Hz narrowband pre-filter. The reported result is that average $\xi<0$7 rises markedly from approximately $\xi<0$8 rad to approximately $\xi<0$9 rad, indicating hypersynchronization during seizure, and that all $f(t)$00 channel-pair DPIs increase significantly with $f(t)$01 (Catanzariti et al., 6 Oct 2025).

The 2D image experiments include both synthetic and real examples. In the simple “face” example, $f(t)$02 denotes the original image, $f(t)$03 a half-intensity version, and $f(t)$04 a structurally modified version. The reported behavior is $f(t)$05 everywhere, while $f(t)$06 and $f(t)$07 show green patches at the modified region. A partition-size study shows that as $f(t)$08 grows from $f(t)$09 to $f(t)$10, localization of the modified region sharpens. In the Van Gogh “Self-Portrait” example, using a grayscale original $f(t)$11, a low-intensity $f(t)$12 at $f(t)$13, and an eye-modified $f(t)$14, a $f(t)$15 partition yields high DPI only in the patches covering the eye, while robustly ignoring the intensity change $f(t)$16.

A separate 2D experiment addresses rotation detection in highly isotropic microscopy. The inputs are a neuron micrograph $f(t)$17 and rotated versions $f(t)$18 with $f(t)$19. The pipeline rotates $f(t)$20 by each candidate $f(t)$21 and compares the result to $f(t)$22 via DPI. The reported result is that the minimizer of DPI correctly recovers the true rotation even for nearly imperceptible angles such as $f(t)$23 and $f(t)$24. Taken together, these experiments position DPI as a single formalism spanning time-series phase divergence, structural image comparison, and rotation-sensitive analysis.

6. Relation to classical phase-difference measures and stated limitations

The paper compares DPI to classical phase-based metrics including phase-lock value, mean phase coherence, and PLI. These classical measures are described as typically relying on 1D Hilbert-extracted phases, being sensitive to amplitude variation unless explicitly normalized, and not being extendable in a straightforward way to images or volumetric data.

Against that baseline, the stated advantages of DPI are that it mathematically extends to arbitrary dimension via Riesz transforms, is invariant to uniform intensity scaling, encodes local geometric structure through sensitivity to edges and textures, and handles rotational transformations by exploiting the rotation-covariance of the Riesz transform. These points specify the sense in which DPI is presented as a generalization rather than merely a new scalar summary of phase disparity.

The limitations are equally explicit. DPI requires Fourier-domain transforms, which are described as costly for large 3D volumes. Phase extraction assumes sufficiently smooth or narrowband content and may therefore require filtering. The pointwise arctangent can be noisy, so local averaging through patches is recommended. These constraints indicate that DPI should not be construed as universally preferable to phase-lock value or PLI; rather, it addresses a different problem class, especially where multidimensional geometry and amplitude-invariant structural comparison are required (Catanzariti et al., 6 Oct 2025).